Professor Manfred Staat ,

Very interesting question indeed. In my opinion the use of strain is more meaningful than stress. I had the same question with regards to the cracking criterion. The criterion is used to find the location of cracking. In the literature, the criteria are mostly based on stress, whether it is the Rankine criterion, Mohr-Coulomb criterion, Drucker-Prager/ Von Mises criteria. Mostly, a combination of shear-based criterion and a tensile criterion is used. I think one of the reasons that strain has been left out of LEFM analyses (and from other fields of mechanical engineering as well) is that the cracking criteria (or failure criteria in general) based on extensional strain were not found successful in estimating the location of the cracking around the crack tip (or failure of material in general). Still, our intuitive understanding is that cracking in opening mode has an extensional nature in essence.

During my PhD research, I tried to look at the problem from a new perspective, discussing the use of a 3D extensional strain-based criterion for opening mode cracking (under low-confinement or tension) and a 3D shear-based criterion for shear mode cracking (under intermediate to high confinement). I discussed this in detail in Chapters 2 and 3 of my dissertation (Directional and 3-D confinement-dependent fracturing, strength and dilation mobilization in brittle rocks (Access: https://dx.doi.org/10.14288/1.0384518)) and explained why the use of a 1-D extensional strain-based criterion that does not consider the influence of confinement on suppression of cracking leads to a wrong estimate of the location of cracking, while use of a 3-D extensional strain-based criterion that considers the influence of confinement on suppression of cracking can give the correct location for crack propagation. I also discussed the application of the criterion to larger scales, including fracturing around tunnels (meso-scale), and regional faults (macro-scale). I hope this would encourage the researchers to re-examine the application of extensional strain-based criteria.

Kind regards,

Masoud

Dear Masoud,

Thank you for your encouraging and detailed answer. In fact, my question grew out of my work on a shear enriched extension strain criterion, which shows to be an alternative presentation of the Mohr-Coulomb criterion with a different failure mode. This phenomenological criterion predicts the "paradox" observation of extension fractures on unloaded material planes in compression tests.

Article An Extension Strain Type Mohr–Coulomb Criterion

I would like to complete the argument with an accepted strain based presentation of fracture mechanics because some people use the stress intensity factor as an argument against strain based failure criteria. I will try to find the time to read your impressive PhD thesis.

Best regards,

Manfred

Dear Professor Saat,

Thank you for your kind compliment on my dissertation.

I have read your paper once it was published. In my work, I took a different approach: I believe in the development of criteria, the possibility of multi- directionality of cracking process should be considered. I started from shear cracking, applied the "multi- directionality of cracking process" concept and found that the limitations of the MC criterion are automatically resolved (including s2 dependency). Next, I extended the applicability of the concept to extensional cracking. And then, merged the two criteria I developed for shear and extensional cracking to derive a 3D spalling - shearing criterion.

In essence, you are suggesting a similar approach in your paper. The difference is that the concept is not applied to the shear cracking process and therefore the validity of the MC criterion is assumed to be valid (i.e., the issue of s2-independnecy of the MC criterion is still there and extends to the enriched criterion).

I am looking forward to hearing your opinion on the approach I took in my work.

Kind regards,

Masoud

Dear Masoud,

My answer is personal. I am a retired professor of mechanics who was not able to explain axial cracking in uniaxial compression to his students. Therefore, I was happy to find the extension strain criterion, although it came too late for my own teaching. But I was surprised how long it has been used and discussed without checking it quantitatively. So I have discussed the shear enrichment, which happens to be the Mohr-Coulomb criterion (MC) but with the difference of extension fractures. So the enriched extension strain criterion could get at least the creditability level of MC.

It was not my intention to improve MC. I have mentioned the shortcomings of MC and have pointed out how the criterion should be used to model the evolution of crack damage. I agree with you that MC can be improved. It was my very first encounter with rock mechanics, and since I am retired, I do not have the opportunity to solve all the problems. I will remain interested, consider other ideas for improving MC, such as yours, and perhaps add something as I find the time to make some progress.

Best regards,

Manfred